Problem: Determine how many solutions exist for the system of equations. ${-9x+3y = 6}$ ${y = 3x+9}$
Solution: Convert both equations to slope-intercept form: ${-9x+3y = 6}$ $-9x{+9x} + 3y = 6{+9x}$ $3y = 6+9x$ $y = 2+3x$ ${y = 3x+2}$ ${y = 3x+9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x+2}$ ${y = 3x+9}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.